Abstract: This paper is dedicated to a description of the poles of the Igusa local zeta function when satisfies a new non-degeneracy condition called arithmetic non-degeneracy. More precisely, we attach to each polynomial a collection of convex sets called the arithmetic Newton polygon of , and introduce the notion of arithmetic non-degeneracy with respect to . If is a -adic field, and is arithmetically non-degenerate, then the poles of can be described explicitly in terms of the equations of the straight segments that form the boundaries of the convex sets . Moreover, the proof of the main result gives an effective procedure for computing .